12 02 2024 E TS

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GAME THEORY - February 12, 2024 ID #: Last name: First name: Part two. Bargaining: explain a non-cooperative approach (describe bargaining as an extensive game, the concept of impatient players, a possible rational solution) and/or a cooperative approach (Nash and/or Kalai- Smorodinski approach, their properties, issues...) Part three. State and prove Nash’s bargaining theorem (write accurately both the hypothesis and the thesis). Exercise 1 Consider the non cooperative game in strategic form, where the entries of the bimatrix represent utilities.   (1, 1) (5 , 2) (11 , 3) (6, 2) (5 , 2) (1 , 6) (3, 5) ( a, 5) (2 , 2)   and assume that Player 2 chooses (1 /2, 1/4, 1/4). What are the best responses of Player 1? Answer of exercise 1 The expected outcomes for the three pure strategies are 18/4, 18/4, (8+a)/4. Then the first and the second strategies are best responses for all a ≤ 10 and the third strategy is a best response for all a ≥ 10. Exercise 2 Consider the non cooperative game in strategic form, where the entries of the bimatrix represent utilities.   (0, 4) (1 , 6) (0 , 3) (4 , 2) (15, 7) (0 , 4) (4 , 6) (3 , 8) (4, 2) (2 , 1) (7 , 4) (0 , 2)   . Assuming that the social utility function is the sum of the utilities of the players, and that only pure strategies are available, what is the price of stability? Answer of exercise 2 The optimal outcome is (15,7), while the only NE in pure strategies is (7,4). Then the price of stability is (15+7)/(7+4)=2. Exercise 3 I have 3 objects, and there are two players, P1 and P2. P1 makes an offer to P2, and her choices are to offer 1, or 2, or 3 objects. P2 can accept the offer, and in this case the game is over. Otherwise, I take off one object and P2 must make a counteroffer of 1 or 2 objects. P1 can…